Assume that $C$ is a negatively oriented, piecewise smooth, closed curve. Let $R$ be the region enclosed by $C$. Use the circulation form of Green's theorem to rewrite $ \iint_R 4x^3y - x^2 \, dA$ as a line integral. Choose 1 answer: Choose 1 answer: (Choice A) A $ \oint_C x^2y \, dx + yx^2 \, dy$ (Choice B) B $ \oint_C -xy \, dx + x^2 \, dx$ (Choice C) C $ \oint_C xy^2 \, dx - xy \, dy$ (Choice D) D $ \oint_C -y^3 \, dx + x^2y^2 \, dy$ (Choice E) E Green's theorem is not necessarily applicable.
Solution: Assume we have a two-dimensional vector field $F(x, y) = P(x, y) \hat{\imath} + Q(x, y) \hat{\jmath}$ and a piecewise smooth, simple, closed curve $C$. Let $R$ be the region enclosed by $C$. Then the circulation form of Green's theorem states that we have the equality below: $ \oint_C P \, dx + Q \, dy = \iint_R \left( \dfrac{\partial Q}{\partial x} - \dfrac{\partial P}{\partial y} \right) dA$ Our first step should be to confirm that the given curve is compatible with using Green's theorem. Looking closely, the curve $C$ does not satisfy all the conditions of Green's theorem: the problem never specifies that $C$ is simple! Because we can't be sure whether or not the curve is simple, Green's theorem is not necessarily applicable.